Moreover, in Subsection REF , we develop a similar analysis for the extremal cases \(s=0\)  and \(s=1\) .
Trivially, the \(s\) -fractional Gagliardo seminorms diverge as \(s\rightarrow 1\) ; but
it is well known, since [1]}, that, after multiplying by \((1-s)\) , they converge to the classical Dirichlet energy. Analogously, by [2]}, the \(s\) -fractional Gagliardo seminorms multiplied by \(s\)  converge, as \(s\rightarrow 0\) , to the squared \(L^2\)  norm. We refer the reader to [3]} for the corresponding \(\Gamma \) -convergence statements in the setting of compactly supported functions.
Here, without developing an asymptotic analysis of our functional as \(s\rightarrow 0^+\)  and \(s\rightarrow 1^-\) , we determine the minimizers of the functionals \({\mathcal {F}}^T_s\)  among two slope functions for \(s=0\)  (corresponding somehow to the case treated in [4]}, [5]}) and \(s=1\)  and we compute their asymptotic energy density.
